Obviously (like with most audio topics) there's probably some subjectivity here but -

I own a pair of AKG K240 studio monitor headphones. I've always liked their sound, and apart from them getting a bit sweaty after prolonged use (3-4 hours) I really have no complaints.

Recently a buddy of mine who's a musician came over and saw I had them and said they were nice headphones for mixing but that if he was just listening to music it wouldn't be his first choice - he'd rather have some EQ'd headphones.

I've seen a similar sentiment on gaming boards in discussions of headphones.

Was thinking of trying them just for kicks but I'm curious if anyone here has any insight they can share? I mean I get what he's saying - but wouldn't monitor headphones really be representing how the music is "supposed" to sound? Wouldn't EQ'd headphones be coloring the sound with the sound of the headphones themselves?

I think we should define what a "monitoring headphone" is and if it has any actual technical meaning that would give it a certain sound.There's an old SOS article comparing monitor vs hi-fi speakers and the conclusion is that there isn't a significant sound difference between them.

I think a flat frequency response and a detailed transparent sound is generally the goal for both. There might be a tendency for monitoring equipment to accentuate the upper mids or highs, to reveal certain "delicate" frequencies. It seems to be the case with some "classics", like the Yamaha NS10 or Sony MDR-V6 for headphones (AFAIK neither of those are very flat in terms of EQ), but I can't say whether it's the rule for all monitoring speakers and headphones.

Shortly, the key point is that a measured flat response for a headphone doesn't translate in a perceived flat response to the ear, for reasons related to outer and mid ear physiology, to the fact that commercial recordings are often mixed and produced having in mind the sound of speakers radiating in typical listening room etc...So headphone designers who want their project to be perceived flat and balanced, instead of targeting for a flat frequency response, target for uneven frequency curves, like for example the "diffuse filed curve" targeted by every Etymotic design, meant to be perceived flat by the listener.

As someone already pointed out, anyway, if the transducers are of good quality, i.e. low distortion in all audio spectrum, equalization can do a lot to suit the sound signature to listener's own taste, be it even and balanced or not, so an answer to the OP question could be: if you're not satisfied by the headphone you currently own or want to try something new, before speeding out to buy a new one, play a little with the graphic equalizer you'll very probabily find as a feature in every player.

There are even some smartphones apps, like Accudio for iOS, that have a database of measurements of the most diffused headphones and try to mimic the sound of many specific models by using the one owned by the user and applying a compensation curve.